Soft Computing manuscript No.(will be inserted by the editor)

Interval Type-2 FuzzyModelling and StochasticSearch for Real-WorldInventory Management

Simon Miller · Mario

Gongora

· Jon Garibaldi ·

Robert John

Received: date / Accepted: date

Abstract Real-world systems present a variety of chal-

lenges to the modeller, not least of which is the problem

of uncertainty inherent in their operation. In this re-search, an Interval Type-2 Fuzzy model is applied to a

real-world problem, the goal being to discover a suitable

optimisation configuration to enable a search for aninventory plan using the model. To this end, a series

of Simulated Annealing configurations and the Interval

Type-2 Fuzzy model were used to search for appropriate

inventory plans for a large-scale real-world problem.A further set of tests were conducted in which the

performance of the Interval Type-2 Fuzzy model was

compared with a corresponding Type-1 Fuzzy model.In these tests the results were inconclusive, though,

as will be discussed there are many ways in which

Type-2 Fuzzy Logic can be exploited to demonstrateits advantages over a Type-1 approach. To conclude,

in this research we have shown that a combination of

Interval Type-2 Fuzzy Logic and Simulated Annealing

is a logical choice for inventory management modellingand inventory plan search, and propose that the bene-

Simon MillerHorizon Digital Economy Research / Intelligent Modellingand Analysis, University of Nottingham, Nottingham, UKTel.: +44 (0)115 8232556E-mail: s.miller@nottingham.ac.uk

Mario Gongora · Robert JohnCentre for Computational Intelligence, De Montfort Univer-sity, Leicester, UKE-mail: m.gongora@dmu.ac.uk,rij@dmu.ac.uk

Jon GaribaldiIntelligent Modelling and Analysis, University of Nottingham,Nottingham, UKE-mail: jmg@cs.nott.ac.uk

fits that a Type-2 model offers, can make it preferable

to a corresponding Type-1 system.

Keywords Interval Type-2 Fuzzy Logic · Real-World

Inventory Management · Simulated Annealing

1 Introduction

Optimising inventory levels within a supply chain is

an area of ongoing interest for supply chain managers.Planning the allocation of resources within a supply

chain has been critical to the success of manufacturers,

warehouses and retailers for many years. Poorly man-

aged resources result in two main problems:

(a) Stock outs, where a node within the chain is unable

to satisfy the demand placed on it.

(b) Surplus stock or carry over, where stock is storedfrom one period to the next.

The consequence of stock outs is lost sales, and po-

tentially lost customers. Surplus stock results in addedcost of storing stock and the possibility of stock losing

value as it becomes obsolete. Holding some surplus

stock is advantageous however; safety stock (stock heldas contingency) can be used in the event of an unex-

pected increase in demand or to cover lost productivity.

Various degrees of uncertainty are present in the dif-

ferent data sources used in supply chain management.

This uncertainty is exacerbated in demand forecastsby applying methods of analysis which have varying

degrees of inherent uncertainty within themselves. Fur-

thermore, other data that is often used in resourceplanning such as transportation and other costs, cus-

tomer satisfaction information, etc. is also uncertain.

Therefore, Fuzzy Logic and especially Type-2 FuzzyLogic (T2FL) are particularly appropriate for this prob-

lem. While traditional (or Type-1 (T1)) Fuzzy Logic

(T1FL) has successfully been used many times for

modelling supply chain operation (e.g., Petrovic et al.(2008) and Aliev et al. (2007)), T2FL has been shown

to offer a better representation of uncertainty on a

number of problems (e.g., Hagras (2004) and Karnikand Mendel (1999)) as it is able to retain more infor-

mation about uncertainty and, unlike T1, represent the

linguistic uncertainty and multiple perceptions of real-world terms. Both of these advantages are particularly

appropriate to the problem of inventory management.

The authors believe that T2FL offers a better method of

representing the uncertainty within a supply chain, andthat its advantages can be beneficial to supply chain

managers. In this research an Interval Type-2 Fuzzy

Logic (IT2FL) model is used, as it benefits from some

2 Simon Miller et al.

of the advantages of T2FL, while incurring considerably

less computation.

The search spaces involved in inventory manage-

ment are often very large even for a relatively simple

problem. As such, it is often not possible to carryout an exhaustive search to find a resource plan; a

more efficient method needs to be selected. To address

this, Simulated Annealing (SA) is evaluated for thetask of finding appropriate resource plans in this work.

Initially, Evolutionary Algorithms (EAs) and hybrid

methods were also considered, however, early testingshowed that these methods were not able to achieve

results comparable to SA, which then this became the

focus of the research. This method is able to search

a large solution space, without the need to evaluate ahigh proportion of the possible solutions. One difficulty

in using SA is the identification of an appropriate

configuration, as it presents a significant configurationspace. Here, a series of SA configurations were evaluated

for this purpose using the IT2FL model mentioned

previously as a cost function.

This paper is structured as follows: Section 2 con-

tains background information on Fuzzy Logic, SA and

related work in the problem domain. Section 3 providesan overview of the real-world test scenario that is

being tackled and Section 4 gives details of how the

chosen method is being employed to this end. Section

5 describes the experiments that have been conducted,along with the results. Finally, Section 6 considers the

conclusions that can be drawn from the work, and what

form future work might take.

2 Preliminaries

This section will provide an introduction to the tech-

niques that have been used in this research, and a look

at related existing work in the literature.

2.1 Fuzzy Logic

Traditional fuzzy logic (Zadeh, 1973), also referred to as

‘Type-1 fuzzy logic’, is based on fuzzy set theory (Zadeh,

1965) which extends classical (or crisp) set theory. Ina crisp set, boundaries are sharp, an element either

belongs to a set or it does not. With fuzzy sets an

element may belong to one or more sets to a varying

degree. In Mendel (2001) a fuzzy set F is described as:

“...a generalization of a crisp set. It is defined ona universe of discourse X and is characterized by

a membership function µF (x) that takes on val-

ues in the interval [0, 1]. A membership function

provides a measure of the degree of similarity of

an element in X to the fuzzy set.”

A common example when considering the difference

between crisp sets and fuzzy sets is that of classify-ing people by height (e.g., Klir et al. (1997)). With

crisp sets we might say that a person over 190cm is

tall, and anyone under this height is short. Figure 1ashows how these crisp sets might look. If a person

is 189cm, this classical set definition would classify

them as short as they are under 190cm, a human

however would probably still call them tall as theywould be almost indistinguishable from someone who

is 190cm tall. Clearly, these two crisp sets are not

adequate for describing whether a person is tall ornot, as the boundary creates a situation where a very

small difference in height can result in a completely

different classification. Fuzzy sets allow us to representthese uncertain boundaries by using varying degrees of

membership, usually in the range [0,1]. In this case,

for example, a person who is over 190cm could belong

to the set tall to the degree 1 and the set short tothe degree 0. Then, if a person who is 189cm is to be

classified, they might be tall to the degree 0.99 and

short to the degree 0.01 (the membership grades do nothave to add up to 1). As we move down the height scale,

individuals would have a lower degree of membership to

the set tall and a greater degree of membership to theset short. Figure 1b shows how the fuzzy sets short and

tall might look.

2.1.1 Type-2 Fuzzy Logic

Later in this section (see Section 2.3) it will be shown

that previous work using fuzzy logic to model inventoryplanning problems has been limited to T1FL (e.g.,

Petrovic et al. (2008), Aliev et al. (2007) and Wang

and Shu (2005)). However T1 fuzzy sets represent thefuzziness of the particular problem using a ‘non-fuzzy’

(or crisp) representation - a number in [0, 1].

As Klir and Folger (1988) point out:

“..it may seem problematical, if not paradox-

ical, that a representation of fuzziness is madeusing membership grades that are themselves

precise real numbers.”

This paradox leads us to consider the role of Type-2

(T2) fuzzy sets as an alternative to the T1 paradigm.

T2 fuzzy sets (Mendel and John, 2002) represent mem-bership grades not as numbers in [0, 1], but as T1 fuzzy

sets.

Mendel and John (2002) state in their definition:

Interval Type-2 Fuzzy Modelling and Stochastic Search for Real-World Inventory Management 3

0.2

0.4

0.6

0.8

1

160 170 180 190 200 210

µ

Height (cm)

Short Tall

(a) Crisp

0.2

0.4

0.6

0.8

1

160 170 180 190 200 210

µ

Height (cm)

Short Tall

(b) T1 Fuzzy

Fig. 1: Crisp and T1 fuzzy sets tall and short

“A type-2 fuzzy set, denoted A, is characterizedby a type-2 membership function µA (x, u) where

x ∈ Jx ⊆ [0, 1].”

and later added (Mendel et al., 2006):

“When all µA (x, u) = 1 then A is an interval

T2 FS.”

This definition provides an extra dimension over

that seen in a T1 fuzzy set, making a T2 fuzzy set 3D.

In a Generalised T2 fuzzy set (GT2), the secondarymembership functions are T1 fuzzy sets. The area in

the primary domain covered by a T2 fuzzy set is called

its ‘footprint of uncertainty’ (FOU), this is the union ofall primary memberships. In an Interval Type-2 (IT2)

fuzzy set, the FOU is representative of the whole set,

as all secondary memberships are equal to 1. Figures2a and 2b show how the IT2 and GT2 fuzzy sets short

and tall might look.

The authors go on to list four sources of uncertainty

that are present in the real-world, that are not repre-sented by T1 fuzzy systems (Mendel and John, 2002):

1. The meanings of words that are used in the an-tecedents and consequents of rules can be uncertain

(words mean different things to different people).

2. Consequents may have a histogram of values as-

sociated with them, especially when knowledge isextracted from a group of experts who do not all

agree.

3. Measurements that activate a T2FL system may benoisy and therefore uncertain.

4. The data that are used to tune the parameters of a

T1FL system may also be noisy

T2 fuzzy sets address these problems by offering

added degrees of freedom that may provide a more

complete model of uncertainty.

In John and Coupland (2007) the authors state that:

“The more imprecise or vague the data, then

type-2 fuzzy sets should offer a significant im-

provement on type-1 fuzzy sets.”

They then suggest that while T1FL can be used

to represent the linguistic uncertainty associated with

words, the perception of words (i.e., what words meanto different people) can be better represented with

T2FL.

T2 fuzzy sets have been widely used in a numberof applications (see John and Coupland (2007) and

Mendel (2007) for examples), and on a number of prob-

lems T2FL has been shown to outperform T1FL (e.g.,

Hagras (2004) and Karnik and Mendel (1999)). T2FLis able to represent uncertainty better by retaining

more information and, unlike T1, is able to represent

linguistic uncertainty and varying perceptions of real-world terms. Both of these advantages have applications

in the type of model presented here. Supply chain

operation contains a great deal of uncertainty which canbenefit from the extra dimension of fuzziness offered by

T2FL. Also, the use of words can provide a more natural

interface for supply chain managers allowing them to

describe numerical information linguistically.

2.2 Optimisation

A wide variety of optimisation techniques have been

used to search for solutions to operations management

problems. This section will provide an overview of SA

as this is the method we have used in this study. Thegoal in this work is to find an inventory plan, not to

optimise the model itself. Some work has been done

regarding the use of optimisation methods to design

4 Simon Miller et al.

0.2

0.4

0.6

0.8

1

160 170 180 190 200 210

u

Height (cm)

Short Tall

(a) IT2 Fuzzy (b) GT2 Fuzzy

Fig. 2: Interval Type 2 and Generalised T2 sets tall and short

T2 Fuzzy sets (e.g., Wagner and Hagras (2007), Sanzet al. (2010), Sanz et al. (2011) and Biglarbegian et al.

(2011)) however, optimising the sets themselves is not

appropriate for this problem, as there is no groundtruth and therefore no basis for comparing one set of

fuzzy sets with another. Also, the sets are constantly

being created and deleted, there is not a fixed set offuzzy sets used in the model. In this work we do not

optimise the sets; we use an IT2 fuzzy model as a cost

function for evaluating potential inventory plans.

2.2.1 Simulated Annealing

SA (Kirkpatrick et al., 1983) is inspired by a real-world

phenomenon, in this case the heating and cooling (an-nealing) of metals to reduce defects. An initial solution

is created, then a neighbouring solution is selected and

compared with it. The probability of the algorithmaccepting the neighbour as the current solution is based

upon a parameter value ‘temp’ (representing the tem-

perature) and the difference in quality between the two

solutions. The higher the value of ‘temp’, and the closerit is to the current solution, the more likely it is that the

algorithm will accept an inferior solution (that is, one

with a lower fitness value). The process is then repeatedusing the selected solution as a starting point. Over the

course of a run ‘temp’ is gradually reduced making it

less likely that an inferior solution will be accepted. SAhas been successfully applied to a number of optimisa-

tion problems. For example, in Garibaldi and Ifeachor

(1999) SA was used to tune a fuzzy Umbilical Acid-Base

assessment model for determining information about anewborn infant’s health, and Tang (2004) conducted a

series of experiments that show SA to be a good choice

for optimising a production/inventory system. Melouk

et al. (2004) and Bouleimen and Lecocq (2003) use SAto solve production scheduling problems.

The SA algorithm used in this research is detailed

in Section 4.3

2.3 Resource Optimisation with Fuzzy Models

A number of research projects have explored the use of

optimisation techniques and T1FL models, to search for

good solutions to manufacturing and inventory prob-lems. The nature of these problems means that there are

no direct methods of calculating an appropriate solu-

tion, and the solution spaces are so large that a method

that evaluates all possible solutions is prohibitive. Toperform a search for ideal solutions in this situation,

we can look to the stochastic search methods considered

for this paper: EAs and SA. In this section we will lookat how EAs and SA have been used in existing research

work.

EAs are a popular method of optimising fuzzy re-

source models. In Aliev et al. (2007) a T1 fuzzy systemwas used with a GA to model a supply chain. The GA

searches for a configuration that maximises profit, while

meeting a target service level (percentage of customerorders satisfied) specified by the user; fuzzy sets were

used to describe costs, returns, production capacities,

storage capacities and forecasts. The proposed fuzzy

method, and a crisp method were compared. The crispsystem was unable to produce a feasible configuration if

the actual demand was lower than the forecast demand.

In contrast, the fuzzy model presented was robust andable to cope with fluctuation in demand and production

capacity with little impact on profitability.

A similar approach was presented by Wang and Shu

(2005). T1FL was used to represent customer demand,

Interval Type-2 Fuzzy Modelling and Stochastic Search for Real-World Inventory Management 5

processing time and delivery reliability; a GA found

order-up-to levels which were used to decide how muchstock should be ordered (what level to ‘order-up-to’)

in the next time period. The system attempted to find

the configuration that incurred the minimum cost. Anoptimism-pessimism index was set by the user and

passed to the system. When optimistic, the model as-

sumed the best case scenario for material response time.A pessimistic attitude presumed the worst. The results

showed that more pessimistic strategies increased the

service level, reducing the sales lost through stock outs,

and incurring higher inventory holding cost as morestock was kept. More optimistic strategies resulted in a

drop in service level and an increase in sales lost, though

the cost of holding inventory was also reduced.Fayad and Petrovic (2005) carried out work using

a T1FL job scheduling model with a GA to find near-

optimal solutions. In this case, a real world example(a printing company) was used to demonstrate the

model. Fuzzy sets were used to describe due dates,

processing times and the level of customer satisfaction.

The experiments described show that the GA was ableto find good solutions.

Earlier work by Sakawa and Mori (1999) also de-

scribed a method of scheduling jobs that have a T1fuzzy due date and processing time. A GA was used to

find schedules. The GA took into account the similarity

of the solutions in a given population. When the initialindividuals were created they had a similarity of 0.8

or less to ensure diversity. It was shown that ensuring

diversity in this way led to the GA finding an optimal

solution on more occasions than a GA without a simi-larity measure. In testing, the GA was shown to work

well, finding solutions with a large correlation between

the processing time and the due date. For comparison, aSA process was used, the GA was shown to consistently

outperform the SA method. Not only were the solutions

better, but the results were more stable. The variancein the best solutions found with SA from test to test

was much larger than that of the GA. The computation

time for the GA was also shown to be faster.

3 Problem Scenario

In this section the problem that was tackled is de-

scribed. The data set was provided by one of theindustrial partners, Unipart Expert Practices Ltd.

The supply chain consists of three tiers, in the

first is a manufacturing distribution centre based in

Newquay, and the second contains distribution centresin Stockport (England), Dublin (Ireland) and Orleans

(France). The final tier contains customers spread over

the UK, Ireland, France and Spain.

Early testing with the model showed that it was not

possible to use the whole test scenario for evaluationpurposes because of the size of the problem. For these

experiments, a subset of the whole problem containing

100 customers and 100 products was chosen, this re-duced problem is still considerably larger than any of

the fuzzy inventory models seen in the literature (see

Table 2). The customers were evenly distributed amongthe three distribution centres in the second tier, and

the products were chosen to represent those stocked at

all three distribution centres. An overview of the test

scenario used can be seen in Table 1.

Table 1: Real-world scenario test supply chain setup

Tiers 3Nodes 1,2,100Products 100Periods 13Batch Cost 100Distance 100Stock out 25MultiplierHolding Cost 10%

purch. priceProduction Cost 1.2Purchase Cost 1.2Transport Cost 0.1

3.1 Cost values

The aim of the research is to find an inventory plan

for this scenario using the IT2 model and SA. In orderto calculate the cost of a potential plan, the following

values are used.

(a) The cost of setting up an order is called the batch

cost. This represents the cost of administration,

setting up any machines that are required, andpicking the items for dispatch. There is a flat fee

for each batch (that can vary by node) which is

charged once at each warehouse for the productionof a particular item for a particular customer.

(b) Products have a production cost. The total produc-

tion cost for each batch is calculated by multiplyingthe number of items by the production cost.

(c) Transport cost represents the cost of carrying an

item between nodes and customers. The product

of the transport cost and distance gives the overalltransport cost for a batch of product.

(d) A holding cost is charged if a product is kept at a

node for more than one period. The cost is calcu-

6 Simon Miller et al.

lated by taking a specified percentage of the pur-

chase price of the goods held, for items carried overfrom one period to the next. The purpose of the

charge is to represent the cost of storing items, the

depreciating value of stock and the losses incurredby tying up capital in unsold stock.

(e) A stock out cost is charged for the shortfall of

a product in a particular period. In this modelwe make the assumption that the end customer

is always provided with an item. If it is not in

the warehouse, it is obtained at purchase price

from a competitor. The stock out cost is the sumof the value of items that had to be purchased.

To discourage discovery of solutions that have a

shortfall of stock, a stock out penalty is added bytaking the product of the stock out cost and a

multiplier provided by the user. Stock out penalty

is applied to all but the final echelon, which suppliesthe end customer. In the final echelon, service level

is used to determine how good a solution is. To

administer a stock out penalty as well would be

to penalise a solution twice for the same shortfall,leading to the SA algorithm being pressured to find

solutions that satisfy 100% of customer demand,

regardless of the service level required by the user.(f) To discourage solutions that do not meet service

level requirements, a service penalty is added to the

cost of poor solutions in proportion to a solution’sdistance from a specified target service level. Ser-

vice level is calculated by taking the percentage of

customer demand that is completely satisfied. To

measure satisfaction the fuzzy sets for the currentstock level and the forecast are compared. An agree-

ment index is calculated by looking at where the sets

intersect, or is set to 1 if the inventory level exceedsthe forecast. As stated before, stock out penalties

are not applied here. It may appear that a more

satisfactory solution would be to simply measureservice level throughout the chain and not use stock

out penalties. However in practice applying service

penalty throughout the model results in SA finding

solutions in which the nodes within the chain placelittle or no orders on each other, enabling solutions

to achieve a good service level without satisfying a

significant amount of customer demand. Stock outsneed to be charged within the chain however, else

SA finds solutions in which only the final echelon

before the customer supplies any product.

3.2 Additional data

It became clear early on in the research that some values

were missing from the data set that had been provided,

and the missing data proved to be unavailable for our

experiments. To remedy this, a number of methodswere agreed with the industrial partner to generate the

data, while maintaining the integrity of the data. The

data did not state which distribution centre suppliedeach customer; as stated previously, customers were

served by one fixed distribution centre. To deduce the

supply relationships, the products being demanded byeach customer were examined. The distribution centre

that supplied the most of these was then assigned to

the customer. Products that appear in the forecast but

were not listed as supplied by any distribution centrewere allocated to the distribution centre of the customer

demanding the product and given a lead time, minimum

order quantity and unit of order that were an averageof all products listed.

For the Newquay and France distribution centres,

there was some information missing regarding lead

times; after discussing this with our industrial partner,the missing information was taken from the Stockport

distribution centre. Capacities, minimum order quanti-

ties and order quantities were not explicitly given in thetest data; to derive these, the forecast was examined,

and the smallest order (capped at 1,000) for a product

at a node was used to represent both the minimum

order quantity and order quantity. For Newquay, theminimum orders and units of order were calculated by

multiplying the Stockport numbers by 35, which is in

line with the demand that it could reasonably expect.The capacities were set to 20 times the minimum order

quantity. A flat batch cost of £100 was applied across

all distribution centres.

3.3 Comparison

Table 2 shows various aspects of the problem being

tackled in this work, the proposed solution, and those

of other projects described in the literature. Thoughthese are all different problems, with different models

(and therefore not directly comparable), the table does

illustrate the difference in size of the problem being

tackled in our research.

4 Proposed System

This section will focus on the model that has been

created and the implementation of the optimisation

methods that are used.

Interval Type-2 Fuzzy Modelling and Stochastic Search for Real-World Inventory Management 7

Table 2: Comparison of problems and models

Petrovic Aliev Wang Proposed(2008) (2007) (2005) System

Tiers 2 3 7 3Nodes 1 4 15 4

Customers 3 2 1 100Products 1 3 6 100Periods - 2 26 13

Fuzzy Logic T1 T1 T1 IT2Optimisation Iterative GA GA SA

4.1 Model

The proposed model represents the interaction of nodes

within a multi-tier supply chain. Figure 3 providesan example of a typical supply chain. In each tier

(or echelon) there are one or more nodes that supply

the subsequent tier with one or more products, andreceive stock from the preceding tier. The first tier

receives goods from an external supplier which for the

purpose of these experiments is assumed to have infinite

capacity, the final tier supplies the customer. Below thefirst tier, capacity is limited by node and product. If

the external supplier were not able to supply 100%

of the first tier’s requirements, the external supplierwould need to become part of the model, allowing us

to represent the limitations of their supply.

A B C C1Raw

Materials

Fig. 3: A typical supply chain

Customer demand is provided by a forecast which is

given to the model at run-time. This forecast represents

the demand placed upon the final tier in the supplychain. Tiers above this can see their own demand

by looking at the suggested inventory levels in the

succeeding tier, as they will be required to supply theseitems.

In order to use the model, the following information

must be provided:

– Number of echelons (not including the end cus-tomer)

– Number of nodes in each echelon

– Number of end customers

– Number of products– Number of periods

– Service level required (as a percentage of orders

filled completely)

– Capacities for each product at each node (amount

that can be produced in one period)– Lead time (in periods) for production/supply of

each product at each node

– Minimum order and unit of order quantities for eachproduct at each node

– Initial stock levels for each node

– Distance between nodes in successive echelons– Forecast of customer demand

– Suggested inventory levels

– Costs including:

– Batch cost– Production cost

– Transport cost

– Holding cost (as a percentage of purchase price)– Purchase price

This information is provided using a series of .csv

files, which are loaded into the model before it isexecuted. As will be shown in Section 4.1.1 some of

these values are represented using IT2 fuzzy numbers

to account for the uncertainty they present. Using thisinformation the model will calculate the cost of the

given resource plan as shown in Equation 1 where F (x)

is the fitness of the solution x and:

– T = total number of periods to be evaluated

– I = total number of inventory locations

– b = batch cost– p = production cost

– tr = transport cost

– h = holding cost

– s = stockout cost– sr = service level

– tsr = target service level

F (x) =

(

T∑

t=0

I∑

i=0

(bi,t + pi,t + tri,t + hi,t + si,t)

)

+max ((tsr − sr) 1000000, 0) (1)

The resource plan is provided by the SA algorithm,

and takes the form shown in Section 4.3.

4.1.1 Interval Type-2 Fuzzy Numbers

In this model, IT2 fuzzy numbers (represented usingIT2 fuzzy sets) are used to represent some values,

and fuzzy arithmetic is used to calculate costs using a

method seen in Kaufmann and Gupta (1985) and Ham-

rawi and Coupland (2009). Fuzzy sets are representedusing a series of α-cuts, each set is an array of pairs

of intervals. Each pair shows the area of values in x

covered at a particular value of µ, the first interval is

8 Simon Miller et al.

the left hand side of the set, and the second the right.

Operations on the IT2 fuzzy sets are performed at theinterval level, corresponding intervals (at the same µ)

are taken from two sets, the operation performed and

the result stored in a third fuzzy set. Four intervallevel operations were required to compute the fitness

of solutions as shown in Equation 1; sum, subtract,

multiply and maximum intersection.

Algorithms 1, 2, and 3 describe the addition, sub-traction and product processes respectively where:

– x = first IT2 fuzzy number

– y = second IT2 fuzzy number

– z = resulting IT2 fuzzy number– xi = ith pair of intervals in x

– yi = ith pair of intervals in y

– zi = ith pair of intervals in z

– at = temporary interval a

– bt = temporary interval b

for each i in I:

zi = [xas

i+ yas

i, xae

i+ yae

i, xbs

i+ ybs

i, xbe

i+ ybe

i]

Algorithm 1: IT2 fuzzy addition

for each i in I:

zi = [xas

i− yae

i, xae

i− yas

i, xbs

i− ybe

i, xbe

i− ybs

i]

Algorithm 2: IT2 fuzzy subtraction

for each i in I:

at = [aas

i× bas

i, aas

i× bae

i, aae

i× bas

i, aae

i× bae

i]

zas

i= min(at)

zae

i= max(at)

bt = [abs

i× bbs

i, abs

i× bbe

i, abe

i× bbs

i, abe

i× bbe

i]

zbsi

= min(bt)

zbei

= max(bt)

Algorithm 3: IT2 fuzzy product

To compute service level, an agreement index be-

tween the forecast and the current inventory level was

calculated using the maximum intersection of the sets

representing inventory and forecast. The agreement

index was then used to calculate service level. Thepseudocode shown in Algorithm 4 describes the process

in detail.

h = 1/I

IF abs

I−1 >= bas

I−1 THEN result = 1, EXIT

ELSE

for each i in I:

IF (abs

i> bas

iAND abs

i< bae

i)

OR (abs

i> bae

iAND abs

i< bas

i+1)

THEN result = (i+ 1)× h,EXIT

Algorithm 4: IT2 fuzzy maximum intersect

Figure 4 shows how the set ‘about 200’ may look

with the α-cut representation used, where x is the scaleof values being represented. Expressing variables in this

way offers supply chain managers the means to state the

uncertainty associated with a particular variable, and

their confidence in their appraisal, this is not possiblewhen using a T1 system without discarding some of the

information.

In order to produce an output that can be applied

to a real-world supply chain, some of the IT2 fuzzynumbers need to be defuzzified. Defuzzification is the

process of taking a fuzzy set and deriving a crisp value

from it. To do this, the Karnik-Mendel method (Karnikand Mendel, 2001) is used. This is a widely used method

that finds an interval representing the centroid of an

IT2 fuzzy set. The interval can then be used to obtain

a crisp number by finding its centre.

In the literature, other than the author’s previouswork (e.g., (Miller and John, 2010)), fuzzy systems that

address inventory management problems have been

restricted to T1FL systems that represent relativelysmall supply chain networks. This work proposes that

such an approach limits the value of a model, and that

T2FL may provide a better solution. IT2FL has beenused because it is computationally cheaper than general

T2FL as it restricts the additional dimension, referred

to as the secondary membership function, to only take

the values 0 or 1. We believe that the extra degree offreedom offered over a T1FL model will allow a better

representation of the uncertain and vague nature of

data used in supply chain management.

Interval Type-2 Fuzzy Modelling and Stochastic Search for Real-World Inventory Management 9

0.4

0.2

0.6

0.8

1

199 200 201 202198 203

µ

x

Fig. 4: Interval representation of IT2 fuzzy set ‘about

200’

All of the data in the real-world scenario is crisp,

to reflect the uncertainty in the data some values were

fuzzified. The following values were represented usingIT2 fuzzy numbers.

– Forecast demand– Inventory level

– Transportation distance

– Transportation cost– Stock out level

– Stock out cost

– Carry over

– Holding cost

For each of these values we can use the linguistic

term ‘about n’, e.g., forecast demand of product a forcustomer b in period c may be ‘about 200’.

The forecast was fuzzified using a primary and

secondary fuzzification factor, a primary fuzzificationfactor of 10% of the value (representing uncertainty)

and a secondary fuzzification factor of 2% of the value

(representing confidence) were chosen for the forecast.

For these experiments the values chosen are somewhatarbitrary, though the primary fuzzification value should

be larger than the secondary fuzzification value. The

primary fuzzification factor creates the principle mem-bership function (which can be thought of as a T1 set

that the IT2 set is based around), and the secondary

fuzzification factor is used to produce the FOU. Largeror smaller values can be chosen to reflect an expert’s

opinion of the margin of error of a forecast and their

confidence in their opinion.

4.2 Inventory Plan Optimisation

The purpose of these experiments is to evaluate the

performance of a set of optimisation configurations.

Initially, EAs, SA and a hybrid method were chosen

to conduct the search for inventory plans as these havebeen shown to work well in previous experiments using

earlier versions of the model, and simpler problems than

the one used in this research. In Miller et al. (2009)experiments were performed with 5 methods: EAs, SA,

the Great Deluge Algorithm (GDA) and hybrids of an

EA and SA and an EA and the GDA. The results of theexperiments showed that SA was able to find the best

inventory plans, but that by combining an EA with SA,

comparable costs could be achieved in less than half the

time. Subsequent work in Miller et al. (2010) showedhow varying the configuration of an EA with variable

mutation and uniform crossover could improve results

gained using the method. In this paper, a large-scalereal-world problem is modelled, presenting additional

problems (e.g., scaling issues) during optimisation. This

work focuses on discovery of an ideal optimisationmethod and configuration for use with such a problem.

In preliminary experiments however, it quickly be-

came apparent that the EA and hybrid algorithm (a

combination of EA and SA) were performing signifi-cantly worse than SA. For this reason, we chose to focus

our work on the SA algorithm.

4.3 Simulated Annealing Implementation

In this implementation, SA consists of the followingcomponents:

(a) The representation used to describe potential re-source plans is a three dimensional matrix of in-

ventory levels. The following pseudocode describes

the array.

float PLAN[Periods][Nodes][Products]

Each element of the matrix contains a floating pointvalue (between 0 and 1) representing the number of

items held in a time period, by a source node of a

particular product. For example if PLAN[1][2][25]

held a value of 0.5, this would indicate that in period1, node 2 is holding 50% of its capacity of product

25. Using the capacity in this way limits the values

that can be created, it is not possible to overloada node. In addition to this values are restricted by

minimum order quantities, and order quantities. For

example, there may be a minimum order quantityof 500 for a particular product, and then orders

restricted to 100s above that (e.g. 500, 600, 700,

800 etc.).

(b) The initial solution is randomly generated, contain-ing only valid inventory levels between the minimum

order quantity and the capacity, in units that com-

ply with valid order quantities.

10 Simon Miller et al.

(c) Fitness is evaluated using the IT2FL model de-

scribed. A solution’s fitness is judged by the costand service level achieved, Equation 1 describes how

this is computed. For those solutions that meet the

service level requirement specified by the user, thefitness is the cost of the solution; for those that do

not a penalty is added to the cost as discussed in

Section 4.1. This encourages SA to find low costsolutions, that meet a target service level.

(d) Selection of a new solution is based on its fitness and

the current temperature. If the perturbed solution is

better than the current solution it is selected, if it isnot, then its distance from the current solution and

the temperature are used to produce a probability

of selection. The closer it is to the current solution,and the higher the temperature, the more likely it

is to be selected.

(e) Perturbation takes a selected solution and thenreplaces a variable amount of elements with other

valid values to create a child. Elements are selected

for replacement at random, the number of changes

is decided using a fitness ranking proportionatemethod similar to roulette wheel selection, the dif-

ference being that the chance of selection is rank

proportionate rather than fitness proportionate. Forexample, in a population of 100, the individual

ranked 1st is twice as likely to be chosen as the

individual ranked 50th. In this case, it means thatit is more likely that a smaller number of pertur-

bations will be chosen, though it is possible that a

large number will be made.

(f) Temperature is reduced after a given number ofiterations with improvements have passed, or if a

number of iterations without improvements (usu-

ally less) have passed. The reduction is made bysubtracting the decrement value from the current

temperature.

(g) Termination of the algorithm occurs when the tem-perature reaches zero.

5 Experimental Methods

Three stages of experiments have been conducted, theseare:

1. Identification of a suitable configuration (Sec-tion 5.1) - The size of the problem prohibits the

running of in-depth tests on a range of configura-

tions. This initial set of experiments aims to discover

a configuration that can be used in further, moreextensive tests, in a reasonable time frame.

2. Performance evaluation (Section 5.2) - Once we

have a suitable configuration, we will perform a

more extensive set of sets to evaluate the perfor-

mance of SA and our IT2FL model on the real-worldproblem that has been selected.

3. Comparison with T1 system (Section 5.3) - The

final set of tests will use the SA configuration foundin the first stage to search for inventory plans using

a corresponding T1 model. The purpose of this stage

is to identify differences in performance between T1and IT2, and decide whether one model can be said

to be better than the other for this problem.

5.1 Identification of a suitable configuration

The first stage of testing aimed to identify an appropri-

ate area of the configuration space that was suitable for

further experimentation. It was shown in Miller et al.(2009) that SA setup is entirely dependent upon the

problem domain, so the temperature and the rate of

decrement were varied. In all tests 100 iterations wereallowed to pass with improvements and 50 iterations

without improvement before the temperature was re-

duced by the decrement value. Table 3 shows the base

configuration used.

Table 3: Base configurations for SA

Its. with improvement Its. no improvement

100 50

In this first set of tests, configurations of SA were

explored; the configurations can be seen in Table 4

along with the results. For these exploratory tests, timewas a factor; only a single test was executed for each

configuration. Early tests showed that runs took a very

long time to complete with the test scenario. It wasalso with time in mind that the configurations have

been chosen, far fewer individuals were evaluated than

in previous work in order to allow these preliminarytests to be completed within a reasonable time frame.

Configurations were chosen to result in a comparable

number of evaluations in each, to avoid a situation

where more and more iterations were performed. Whena suitable configuration was discovered for this problem,

longer tests were executed to take advantage of multiple

runs.

5.1.1 Results

The best results came from in test 3 when a starting

temperature of 25,000 was used, Table 5 shows the

attributes of the plan found.

Interval Type-2 Fuzzy Modelling and Stochastic Search for Real-World Inventory Management 11

Table 4: Results of first stage of testing

No. Temp. Dec. Cost

1 100,000 1,000 £608,914,0482 50,000 500 £601,184,9603 25,000 250 £598,187,8404 12,500 125 £602,834,6885 125,000 1250 £606,111,1046 150,000 1500 £610,011,6487 10,000 100 £619,027,0088 15,000 150 £599,733,6329 20,000 200 £612,790,46410 30,000 300 £606,846,08011 40,000 400 £604,292,35212 60,000 600 £604,731,00813 70,000 700 £600,298,88014 80,000 800 £608,938,36815 90,000 900 £619,253,76016 21,000 210 £607,766,65617 22,000 220 £606,997,95218 23,000 230 £600,488,57619 24,000 240 £601,860,60820 26,000 260 £600,713,98421 27,000 270 £606,076,09622 28,000 280 £618,071,10423 29,000 290 £609,204,032

It was promising that less than 1% of the total cost

was incurred through stockouts. As should be expected,

the majority of the cost was production/procurementcosts. Holding cost, however, was a concern, in this

plan as 29.4% of the total cost came from holding stock

in warehouses. This suggests that too much stock wasbeing allocated to the warehouses. As stated before,

this can result in products losing value and means that

capital is unnecessarily tied up in stock. For seasonalproducts, it can mean that stock is being held while it

is out of season.

Table 5: Test 3 - Resource plan attributes

Total Cost Service Level Batch Cost

£598,187,840 99.9% £9,067,901 (1.5%)

Production Cost Transport Cost

£407,605,024 (68.1%) £899,522.19 (0.2%)

Stockout Cost Holding Cost

£4,886,310.50 (0.8%) £175,729,056 (29.4%)

In all subsequent tests in this stage, the result

produced using the configuration in test 3 was neverbettered. This is the configuration that was used in

the next stage of testing for a more extensive look at

performance.

5.2 Performance Evaluation

With the best configuration found in the first stage,

extra tests were completed. Table 6 shows the setup

used, where stalls refers to the number of iterations thatwere allowed to pass without improvement. In all, 30

tests were run with differing random seeds. The purpose

of these tests was to give a more complete picture ofthe performance of the model and optimisation with

the settings found, enabling a comparison with a cor-

responding T1FL system in the next stage of testing.Table 7 provides a summary of the results.

5.2.1 Results

The results showed that there was a spread of

£40,183,616 between the most expensive and cheapest

solutions found. In real terms this represents a signif-

icant amount, though, relative to the costs themselvesthe best is just 6.4% cheaper than the most expensive;

along with the standard deviation, this suggests that

that SA is finding relatively stable solutions. In the nextstage of testing, a corresponding T1 fuzzy system was

compared with the IT2 system to see if there was any

difference in the results, and whether one system couldbe said to be better than the other.

Table 6: Optimisation setup for second stage of testing

Temp. Dec. Iterations Stalls

25,000 250 100 50

Table 7: Summary of results of second stage of testing

Min. Cost Max. Cost Mean Cost

£586,283,904 £626,467,520 £605,181,126.4

Std. Dev.

£10,110,866.88

5.3 Comparison with T1 System

In this stage a T1 system was created that represented

how the IT2 system would be, if it were implementedusing T1FL. To create a T1 system, the intervals in

the IT2 system were reduced to zero width eliminating

their FOUs. Both IT2 and T1 tests use the same model,

12 Simon Miller et al.

because of this, their execution times are the same as

the computation is not reduced for the T1 version of themodel. The tests performed in Stage 2 were run again.

The purpose of these tests was to see the difference

made by using IT2 over T1, and whether it could bedemonstrated that one approach worked better than

the other. Table 8 provides a summary of the results.

5.3.1 Results

The results appeared to be similar on first inspection.

The cost of solutions found by the T1 system were

marginally more expensive, and had slightly more de-viation. A range of £43,756,032 existed between the

cheapest and most expensive solutions found; the best

was 7% cheaper than the worst. The attributes of theresource plans found in the best of the T1 and IT2 tests

are provided in Table 9, in both cases 99.9% of customer

demand was satisfied.

Table 8: Summary of results of third stage of testing

Min. Cost Max. Cost Mean Cost

£584,430,912 £628,186,944 £606,121,892.3

Std. Dev.

£11,316,239.36

Table 9: Best resource plan attributes found in thirdstage of testing

Type Total Cost Batch Cost

T1 £584,430,912.00 £9,067,501.00 (1.6%)IT2 £586,283,904.00 £9,067,699.00 (1.5%)

Type Production Cost Transport Cost

T1 £396,506,880.00 (67.8%) £899,482.50 (0.2%)IT2 £400,246,080.00 (68.3%) £899,502.25 (0.2%)

Type Stockout Cost Holding Cost

T1 £4,874,550.50 (0.8%) £173,082,480.00 (29.6%)IT2 £5,190,420.00 (0.9%) £170,880,224.00 (29.1%)

Analysing the best solutions it is apparent that

the solution found using the T1 model saved money

by achieving lower production and stockout costs, but

incurred higher holding costs. The best solution wasfound using T1 fuzzy logic, however, on average the

IT2 solutions were slightly better, and more stable.

We can speculate as to the reason for the difference

in results, for example, a likely explanation seems to be

that the extra uncertainty afforded by the IT2 modelleads to a smoother cost function, this could potentially

result in less variation between results, as the ‘steps’

are smaller. However, the resulting cost of a solutionis not necessarily an accurate guide to the quality of a

solution. It may be possible to produce a model that

results in the discovery of a cheaper solution; however,if these solutions do not produce the same results in the

real-world then the quality of the solution is in doubt.

The next step was to examine the results to as-

certain whether there were any significant differences

between the T1 and IT2 tests. Table 10 shows theabsolute value of the mean difference between tests,

the standard deviation of differences, the maximum

and minimum differences (where the IT2 results aresubtracted from the T1 results), and the test statistic

and two-tailed p values produced by performing a t-

test on the two sets of results with α=0.05. The t-test is used to test the hypothesis that there is no

difference between the means of two samples of data.

A test statistic and a p value are produced that can be

used to determine whether the mean difference betweentwo samples of data is statistically significant. Typically

(and in this case) a p value of less than 0.05 is taken

as evidence of there being a statistically significantdifference between the two sets of data.

Table 10: Comparison of results in T1 and IT2 tests

Abs. Mean Diff. Std. Dev. Min. Diff.

£940,765.87 14,885,709.55 £-27,818,496.00

Max. Diff. Test Statistic Test Critical Two Tail

£31,395,136.00 0.34 2.05

Two Tailed p

0.74

The Test Critical Two Tail column shows the value

that the test statistic must be greater than in order

to be statistically significant. In this test it was shownthat neither a p value of less than 0.05, nor a test

statistic greater than 2.05 (test critical) were produced.

From this, it was deduced that there was no appre-ciable difference between the mean results of the T1

and T2 tests. When considering reasons for this we

should note that the IT2 fuzzy sets were created in a

symmetrical manner, that is, by taking a T1 set andshifting the boundaries. It may be that in this case

the approach taken has resulted in an IT2FL system

that is roughly equivalent to the T1FL system it was

Interval Type-2 Fuzzy Modelling and Stochastic Search for Real-World Inventory Management 13

compared with, reducing the advantage IT2FL can

provide. To tackle this shortcoming, an approach thataims to model the actual uncertainty in the data (which

is significantly more complex), rather than applying

a uniform uncertainty should be adopted. By doingthis, the extra information that the model represents

would be based only on actual information provided

by the user, providing a more accurate description ofuncertainty.

One possible view is that this is a positive result

with regard to the fact that the IT2FL model has

the advantages that IT2FL brings, while matching theperformance of a T1 model. In Section 6, the meaning

of these results is considered in more detail, and courses

of action for fully exploiting the advantages of T2FL areproposed.

6 Conclusion

A large-scale real-world case study was collected from aUK supply chain design and management consultancy,

and a series of experiments was conducted to discover

a suitable method and configuration to determine near-optimal resource plans using an IT2 fuzzy supply chain

model. The tests showed that using the model, SA was

able to find realistic solutions to the real-world scenario,satisfying 99.9% of customer demand. As the algorithm

was guided purely by the model, this suggests that the

IT2FL model is a valid representation of the problem.

If this were not the case, they may have been guidedtowards solutions that were unfeasible. However, all of

the tests gave results with high holding costs, showing

that there are some limitations that could be consideredin additional research. The size of the problem tackled

posed problems for the algorithm used, this led to the

algorithm being restricted in order to allow them to runin a reasonable time, on the hardware available, limiting

performance. Future work could look at how resource

plans could be encoded to reduce the resources required

to process them.In an extra set of tests comparing the IT2 system

with a corresponding T1 system it was demonstrated

that there was no statistically significant difference inthe results produced. On reflection, perhaps this is not

surprising. Uncertainty was applied in a symmetrical,

uniform manner; in reality, the uncertainties involvedare much more varied and complex. To address this

IT2FL (or T2FL) should be used in a way that takes

advantage of its strengths. Using an IT2 approach to

this problem allowed the user to specify both the un-certainty of supply chain variables, and their confidence

in their assessment. This was then represented without

compromising the information gathered, unlike a T1

approach. Further to this, a generalised T2 method-

ology could allow the representation of the uncertaintyand confidence levels associated with a group of experts

opinions (as seen in Wagner and Hagras (2010)) while

preserving all of the information gathered includingwhere there was agreement, something not possible

in an IT2 approach. Although some work has been

done here, further exploiting these advantages couldform an important part of future work with the model.

These improvements would thoroughly demonstrate the

advantage of the method chosen over using T1FL, as

correctly modelling words and maintaining all informa-tion from a group of experts, can only be achieved at

this level using T2FL. This makes T2FL a logical choice

when modelling systems that contain a great deal ofuncertainty, as we can produce a more complete model

of the uncertainty present.

Acknowledgements The research reported here has beenfunded by the Technology Strategy Board (Grant No. H0254E).

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